Black brane

 ( Black Holes ) 

In general relativity, a black brane is a solution of the Einstein field equations that generalizes a black hole solution but it is also extended—and translationally symmetric—in additional spatial dimensions. That type of solution would be called a black -brane.

In string theory, the term black brane describes a group of D-brane that are surrounded by a horizon.

With the notion of a horizon in mind as well as identifying points as zero-branes, a generalization of a black hole is a black p-brane. However, many physicists tend to define a black brane separate from a black hole, making the distinction that the singularity of a black brane is not a point like a black hole, but instead a higher dimensional object.

A BPS black brane is similar to a BPS black hole. They both have electric charges. Some BPS black branes have magnetic charges.

The metric for a black -brane in a -dimensional spacetime is: $$\{d\; s\}^\{2\}\; =\; \backslash left(\; \backslash eta\_\{ab\}\; +\; \backslash frac\{r\_s^\{n-p-3\}\}\{r^\{n-p-3\}\}\; u\_a\; u\_b\; \backslash right)\; d\; \backslash sigma^a\; d\; \backslash sigma^b\; +\; \backslash left(1-\backslash frac\{r\_s^\{n-p-3\}\}\{r^\{n-p-3\}\}\backslash right)^\{-1\}\; dr^2\; +\; r^2\; d\; \backslash Omega^2\_\{n-p-2\}$$ where:

• is the -Minkowski metric with Metric signature ,
• are the coordinates for the worldsheet of the black -brane,
• is its four-velocity,
• is the radial coordinate,
• is the metric for a -sphere, surrounding the brane.

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Curvatures When $$ds^2=g\_\{\backslash mu\backslash nu\}dx^\backslash mu\; dx^\backslash nu\; +\; d\backslash Omega\_\{n+1\},$$ the Ricci Tensor becomes $$\backslash begin\{align\}$$

 R_{\mu\nu} &= R_{\mu\nu}^{(0)} + \frac{n+1}{r}\Gamma^r_{\mu\nu}, \\
 R_{ij} &= \delta_{ij} g_{ii} \left(\frac{n}{r^2}(1-g^{rr}) - \frac{1}{r}(\partial_{\mu} + \Gamma^\nu_{\nu\mu})g^{\mu r}\right),
     

\end{align} and the Ricci Scalar becomes $$R\; =\; R^\{(0)\}\; +\; \backslash frac\{n+1\}\{r\}g^\{\backslash mu\backslash nu\}\backslash Gamma^r\_\{\backslash mu\backslash nu\}\; +\; \backslash frac\{n(n+1)\}\{r^2\}(1-g^\{rr\})\; -\; \backslash frac\{n+1\}\{r\}(\backslash partial\_\backslash mu\; g^\{\backslash mu\; r\}\; +\; \backslash Gamma^\backslash nu\_\{\backslash nu\backslash mu\}g^\{\backslash mu\; r\}),$$ where $R\_\{\backslash mu\backslash nu\}^\{(0)\}$, $R^\{(0)\}$ are the Ricci Tensor and Ricci scalar of the metric $ds^2=g\_\{\backslash mu\backslash nu\}dx^\backslash mu\; dx^\backslash nu.$

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Black string A black string is a higher () generalization of a black hole in which the event horizon is topologically equivalent to and spacetime is asymptotically .

Perturbations of black string solutions were found to be unstable for (the length around ) greater than some threshold . The full non-linear evolution of a black string beyond this threshold might result in a black string breaking up into separate black holes which would coalesce into a single black hole. This scenario seems unlikely because it was realized a black string could not pinch off in finite time, shrinking to a point and then evolving to some Kaluza–Klein black hole. When perturbed, the black string would settle into a stable, static non-uniform black string state.

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Kaluza–Klein black hole A Kaluza–Klein black hole is a black brane (generalisation of a black hole) in asymptotically flat Kaluza–Klein space, i.e. higher-dimensional spacetime with compact dimensions. They may also be called KK black holes.Obers (2009), p. 212–213

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See also

• AdS black hole

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Bibliography

• N.A. Obers (2025). 9783540884590 ISBN 9783540884590

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